We consider the quantum hydrodynamic system on a $ d $-dimensional irrational torus with $ d = 2, 3 $. We discuss the behaviour, over a "non-trivial" time interval, of the $ H^s $-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form $ \varepsilon $-small initial conditions, remain bounded in $ H^s $ for a time scale of order $ O(\varepsilon^{-1-1/(d-1)+}) $, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevents to have "good separation'' properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.
Citation: Roberto Feola, Felice Iandoli, Federico Murgante. Long-time stability of the quantum hydrodynamic system on irrational tori[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022023
We consider the quantum hydrodynamic system on a $ d $-dimensional irrational torus with $ d = 2, 3 $. We discuss the behaviour, over a "non-trivial" time interval, of the $ H^s $-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form $ \varepsilon $-small initial conditions, remain bounded in $ H^s $ for a time scale of order $ O(\varepsilon^{-1-1/(d-1)+}) $, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevents to have "good separation'' properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.
[1] | P. Antonelli, L. E. Hientzsch, P. Marcati, Analysis of acoustic oscillations for a class of hydrodynamic systems describing quantum fluids, 2020, arXiv: 2011.13435. |
[2] | P. Antonelli, L. E. Hientzsch, P. Marcati, H. Zheng, On some results for quantum hydrodynamical models, In: Mathematical analysis in fluid and gas dynamics, RIMS Publishing, 107–129. |
[3] | P. Antonelli, P. Marcati, On the finite energy weak solutions to a system in Quantum Fluid Dynamics, Commun. Math. Phys., 287 (2009), 657–686. doi: 10.1007/s00220-008-0632-0 |
[4] | C. Audiard, B. Haspot, Global well-posedness of the Euler–Korteweg system for small irrotational data, Commun. Math. Phys., 351 (2017), 201–247. doi: 10.1007/s00220-017-2843-8 |
[5] | D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys., 234 (2003), 253–285. doi: 10.1007/s00220-002-0774-4 |
[6] | D. Bambusi, J. M. Delort, B. Grébert, J. Szeftel, Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Commun. Pure Appl. Math., 60 (2007), 1665–1690. doi: 10.1002/cpa.20181 |
[7] | D. Bambusi, B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507–567. |
[8] | D. Bambusi, B. Langella, R. Montalto, On the spectrum of the Schrödinger operator on $\mathbb{T}^d$: a normal form approach, Commun. Part. Diff. Eq., 45 (2020), 303–320. doi: 10.1080/03605302.2019.1670677 |
[9] | D. Bambusi, B. Langella, R. Montalto, Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori, 2020, arXiv: 2012.02654. |
[10] | D. Bambusi, B. Langella, R. Montalto, Spectral asymptotics of all the eigenvalues of Schrödinger operators on flat tori, 2020, arXiv: 2007.07865v2. |
[11] | S. Benzoni-Gavage, R. Danchin, S. Descombes, On the well-posedness for the Euler-Korteweg model in several space dimensions, Indiana U. Math. J., 56 (2007), 1499–1579. doi: 10.1512/iumj.2007.56.2974 |
[12] | J. Bernier, R. Feola, B. Grébert, F. Iandoli, Long-time existence for semi-linear beam equations on irrational tori, J. Dyn. Diff. Equat., 2021, 10.1007/s10884-021-09959-3. |
[13] | M. Berti, A. Maspero, F. Murgante, Local well posedness of the Euler-Korteweg equations on $\mathbb{T}^{d}$, J. Dyn. Diff. Equat., 2021, 10.1007/s10884-020-09927-3. |
[14] | M. Berti, J. M. Delort, Almost global solutions of capillary-gravity water waves equations on the circle, UMI Lecture Notes, 2017. |
[15] | J. M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus, JAMA, 107 (2009), 161–194. doi: 10.1007/s11854-009-0007-2 |
[16] | E. Faou, L. Gauckler, C. Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Commun. Part. Diff. Eq., 38 (2013), 1123–1140. doi: 10.1080/03605302.2013.785562 |
[17] | R. Feola, B. Grébert, F. Iandoli, Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori, 2020 arXiv: 2009.07553. |
[18] | R. Feola, F. Iandoli, Local well-posedness for the Hamiltonian quasi-linear Schrödinger equation on tori, 2020, arXiv: 2003.04815. |
[19] | R. Feola, F. Iandoli, Long time existence for fully nonlinear NLS with small Cauchy data on the circle. Ann. Scuola Norm. Sci., 22 (2021), 109–182. |
[20] | R. Feola, F. Iandoli, A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle, 2020, arXiv: 2002.12448. |
[21] | R. Feola, R. Montalto, Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori, 2021, arXiv: 2103.10162. |
[22] | S. Gustafson, K. Nakanishi, T. P. Tsai, Scattering for the Gross-Pitaevskiiequation, Math. Res. Lett., 13 (2006), 273–285. doi: 10.4310/MRL.2006.v13.n2.a8 |
[23] | A. D. Ionescu, F. Pusateri, Long-time existence for multi-dimensional periodic water waves, Geom. Funct. Anal., 29 (2019), 811–870. doi: 10.1007/s00039-019-00490-8 |
[24] | E. Madelung, Quanten theorie in Hydrodynamischer Form, Z. Physik, 40 (1927), 322–326. doi: 10.1007/BF01400372 |
[25] | J. Moser, A rapidly convergent iteration method and non-linear partial differential equations – I, Ann. Scuola Norm. Sci., 20 (1966), 265–315. |
[26] | C. Procesi, M. Procesi, Reducible quasi-periodic solutions of the non linear Schrödinger equation, Boll. Unione Mat. Ital., 9 (2016), 189–236. doi: 10.1007/s40574-016-0066-0 |